D₂₁⋊C₈ #66556

order	336
isoclinism	isoclinic class of D₃×D₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

D₂₁⋊C₈

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₃×D₇
commutator, hypocenter	C₃×C₇	C₂×C₈
Fitting, cosocle	C₃×C₄×C₇	C₂²
socle	C₂×C₃×C₇	C₂×C₄
prime radical	C₂×D₂₁	C₄
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₄

44 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 168

A minimal faithful permutation representation — 2 generators on 18 points:

(1,2,4,6,7,5,3)(9,10)(11,13,14,18,15,12,16,17);(2,3)(4,5)(6,7)(8,9,10)(11,12,14,17,15,13,16,18)

Polycyclic presentation:

2^4,3,7|a^2=c,b^2=c,c^2=d,f^a=f^6,e^b=e^2

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 25);